What make/model of power supply are you using? Assuming you have sufficient power, that GPU will work in your system albeit a bit of overkill when compared to the rest of your rig. Your CPU will likely bottleneck the new GPU.
Your card will work with the system since the connection you will be using is still PCIex16 2.0. I used to have a GTX 560Ti on a Asus P5Q motherboard and a Core 2 Quad Q8200 with 4GB of DDR2 RAM. All of which worked flawlessly no hitch whatsoever.
However you wont be able to get the full capacity of your HD 6970. Your CPU will bottleneck your performance due to games and other applications may require a higher capacity CPU. For example my current build has a i7 920 at the same clocks as the C2Quad Q8200 assassin's creed brotherhood runs at 70FPS compared to 30FPS. I agree with amuffin the only question is would your PSU handle the load of the HD 6970.
Even when I overclocked my Q8200 to 3.8Ghz and my 560Ti to 980mhz but for CPU intensive games like Assassins Creed it was only up to 30FPS then when i changed to an i7-920 the FPS went to an average of 70 FPS. So there is less of a bottle neck (for most games) but there are certain games (not optimized for PC more for consoles) that no matter how much you OC it wouldn't matter.
Like what geekapproved said you wont see the full potential but you will see an improvement over your current setup in most games (but when the scene needs CPU power well there will be a slow down then boost up again).
Finally the new card works ok in the old PC but because I whanted to have the 100% of card's capacity, I decided to buy everything new!!!!!
So I bought:
CPU i7 2600K 3.4 GHz
DDR3 16 GB 2133 MHz
I sold my old PC in good price, and now everything is OK!!!!